Braid Groups and Left Distributive Operations

An extension of the infinite braid group B∞ exactly describes the left distributivity identities. These identities form a decidable family. The group B∞ is closely connected with the free left distributive structure with one generator, and inherits some order properties with simple topological meaning. A quick comparison algorithm for braid words is also given. The first aim of this paper is to construct a proof of the following result which had been conjectured for several years Theorem.There is an effective algorithm for deciding whether a given identity is or not a consequence of the left distributivity identity x(yz) = (xy)(xz). The former status of this question was unusual. Two partial solutions have been proposed independently in [5] and [20] by reducing the decidability to a (unique) algebraic hypothesis known as the Irreflexivity Conjecture. This conjecture has been shown by Richard Laver to follow from a very strong (hence unprovable) set theoretical axiom. The question was whether this additional assumption is necessary or not. The opinions thereabout were divided: on one hand the connection between the existence of very large cardinals and a purely finitistic problem like the one above seemed strange, but on the other hand no metamathematical reason is known to forbid such a connection and some works on distributive structures ([9]) showed that intrinsically complex objects necessarily arise in their description. In this paper we definitely eliminate all logical assumptions in this question and give a solution which is purely algebraic in methods and spirit. We first introduce an algebraic structure on left distributivity identities themselves. Definitions can be made in such a way that the structure CLD thus constructed resembles a group. Then one replaces the study of CLD, which has only a partial product and is incompletely